Matakuliah
Tahun
: MATRIX ALGEBRA FOR STATISTICS
: 2009
EIGENVALUES, EIGENVECTORS
Pertemuan 6
A is an mxm matrix, then any scalar 
satisfying the equation AX = X,
for some mx1 vector X  0, is called an
eigenvalue of A.
The vector X is called an eigenvector of
A corresponding to the eigenvalue A, and
equation AX = X is called the
eigenvalue-eigenvector equation of A.
Bina Nusantara University
Eigenvalues and eigenvectors are
also sometimes referred to as latent
roots and vectors or characteristic
roots and vectors.
AX = X dapat diubah menjadi
(A - I)X = 0
Bina Nusantara University
4
Eigenvalue A must satisfy (A - I) = 0,
called characteristic equation of A
There are scalars α0, ... , αm-1 such that the
characteristic equation above can be
expressed
α0+α1(-)+ ... +αm-1(-)m-1+(-)m
Bina Nusantara University
5
Since an mth degree polynomial has
m roots, it follows that
an mxm matrix has m eigenvalues;
that is, there are m scalars
1, ... , m, which satisfy the
characteristic equation
Bina Nusantara University
6
Contoh:
Find the eigenvalues and eigenvectors of
matrix A
The characteristic equation of A is
Bina Nusantara University
7
= -(5 - ) 2 (2 +) - 3(4) 2 - 4(3) 2
+ 3(4)(2 +) + 3(4)(5 -) + 3(4)(5 - )
= - 3 + 8 2 - 17 + 10
= -(- 5)(- 2)(- 1) = 0
The three eigenvalues of A are 1, 2, and 5
Bina Nusantara University
8
To find the eigenvector
For  =1, solve the equation Ax = 1x for x,
which yields the system of equations
5x1 - 3X2 + 3X3 = x1
4x1 - 2x2 + 3X3 = X2
4x1 - 4X2 + 5X3 = X3
The eigenvector for eigenvalue 1 is
Find the eigenvector for eigenvalue 2 and 5!
Bina Nusantara University
9
Application:
• Covariance matrix
• Multivariate analyses
120)
Bina Nusantara University
(lihat buku 2 hal. 100-
10